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Ukrainian Mathematical Journal

, Volume 57, Issue 9, pp 1522–1534 | Cite as

On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes

  • M. I. Portenko
Article
  • 23 Downloads

Abstract

We construct a Wiener process on a plane with semipermeable membrane located on a fixed circle and acting in the normal direction. The construction method takes into account the symmetry properties of both the circle and the Wiener process. For this reason, the method is reduced to the perturbation of a Bessel process by a drift coefficient that has the type of a δ-function concentrated at a point. This leads to a pair of renewal equations, using which we determine the transition probability of the radial part of the required process.

Keywords

Diffusion Process Normal Direction Symmetry Property Wiener Process Construction Method 
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REFERENCES

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    M. I. Portenko, Diffusion Processes in Media with Membranes [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (1995).Google Scholar
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    K. Ito and H. R. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin (1965).Google Scholar
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    D. S. Kuznetsov, Special Functions [in Russian], Vysshaya Shkola, Moscow (1965).Google Scholar
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    W. Feller, Introduction to Probability Theory and Its Applications, Wiley, New York (1970).Google Scholar
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    A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (1964).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • M. I. Portenko
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKyivUkraine

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